In Chapters 1 and 2, we introduce the definition of interpolating refinable function vectors in dimension one and high dimensions, characterize such interpolating refinable function vectors in terms of their masks, and derive their sum rule structure explicitly. We study biorthogonal refinable function vectors from interpolating refinable function vectors. We also study the symmetry property of an interpolating refinable function vector and characterize a symmetric interpolating refinable function vector in any dimension with respect to certain symmetry group in terms of its mask. Examples of interpolating refinable function vectors with some desirable properties, such as orthogonality, symmetry, compact support, and so on, are constructed according to our characterization results.
In Chapters 3 and 4, we turn to the study of general matrix extension problems with symmetry for the construction of orthogonal and biorthogonal multiwavelets. We give characterization theorems and develop step-by-step algorithms for matrix extension with symmetry. To illustrate our results, we apply our algorithms to several examples of interpolating refinable function vectors with orthogonality or biorthogonality obtained in Chapter 1.
In Chapter 5, we discuss some possible future research topics on the subjects of matrix extension with symmetry in high dimensions and frequency-based non-stationary tight wavelet frames with directionality. We demonstrate that one can construct a frequency-based tight wavelet frame with symmetry and show that directional analysis can be easily achieved under the framework of tight wavelet frames. Potential applications and research directions of such tight wavelet frames with directionality are discussed. / Applied Mathematics
| Identifer | oai:union.ndltd.org:LACETR/oai:collectionscanada.gc.ca:AEU.10048/1224 |
| Date | 11 1900 |
| Creators | Zhuang, Xiaosheng |
| Contributors | Bin Han (Mathematical and Statistical Sciences), Bin Han (Mathematical and Statistical Sciences), John C. Bowman (Mathematical and Statistical Sciences), Rong-Qing Jia (Mathematical and Statistical Sciences), Yau Shu Wong (Mathematical and Statistical Sciences), Mrinal Mandal (Electrical and Computer Engineering), Ding-Xuan Zhou (Mathematics, City University of Hongkong) |
| Source Sets | Library and Archives Canada ETDs Repository / Centre d'archives des thèses électroniques de Bibliothèque et Archives Canada |
| Language | English |
| Detected Language | English |
| Type | Thesis |
| Format | 5762179 bytes, application/pdf |
| Relation | Bin Han, Son-Geol Kwon and Xiaosheng Zhuang, Generalized Interpolating Refinable Function Vectors, Journal of Computational and Applied Mathematics, Vol. 227 (2009), 254--270., Bin Han and Xiaosheng Zhuang Analysis and Construction of Multivariate Interpoalting Refinable Function Vectors, Acta Applicandae Mathematicae, Vol. 107 (2009), No. 1-3, 143--171., Bin Han and Xiaosheng Zhuang, Matrix Extension with Symmetry and Its Application to Filter Banks, SIAM Journal on Mathematial Analsyis, (2010), 20 pages, accepted for publication, Xiaosheng Zhuang, Matrix Extension with Symmetry and Construction of Biorthogonal Multiwavelets, arXiv:1006.2412 |
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